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    Let F be a field and let K be an algebraically closed field with F â?? K.

    If f â??F[x] is irreducible (i.e. if f = m * n , then one of m or n is a constant) and f

    has a multiple zero in K ,

    then f â?² = 0

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    https://brainmass.com/math/basic-algebra/proposition-algebraically-closed-field-414930

    Solution Preview

    Proposition. Let F be a field and let K be an algebraically closed field with F⊆K. If f∈F[x] is irreducible and f has a multiple zero in K, then f^'=0.

    Proof: Before we prove this proposition, let's introduce a theorem related to separable polynomials and their derivatives. A polynomial f∈F[x] is said to be separable if it has distinct zeros in a splitting field over F. That is, each zero of f has multiplicity 1. We now introduce the following theorem.

    Theorem 1. A nonzero polynomial f in F[x] is separable if and only if it's relatively ...

    Solution Summary

    Proposition is clarified here for an algebraically closed field.

    $2.49

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